# Cosine of Complement equals Sine

## Theorem

$\map \cos {\dfrac \pi 2 - \theta} = \sin \theta$

where $\cos$ and $\sin$ are cosine and sine respectively.

That is, the sine of an angle is the cosine of its complement.

## Proof 1

 $\ds \map \cos {\frac \pi 2 - \theta}$ $=$ $\ds \cos \frac \pi 2 \cos \theta + \sin \frac \pi 2 \sin \theta$ Cosine of Difference $\ds$ $=$ $\ds 0 \times \cos \theta + 1 \times \sin \theta$ Cosine of Right Angle and Sine of Right Angle $\ds$ $=$ $\ds \sin \theta$

$\blacksquare$

## Proof 2

 $\ds \map \cos {\frac \pi 2 - \theta}$ $=$ $\ds \map \cos {\theta - \frac \pi 2}$ Cosine Function is Even $\ds$ $=$ $\ds \map \sin {\theta - \frac \pi 2 + \frac \pi 2}$ Sine of Angle plus Right Angle $\ds$ $=$ $\ds \sin \theta$

$\blacksquare$

## Proof 3

 $\ds \map \cos {\dfrac \pi 2 - \theta}$ $=$ $\ds \frac 1 2 \paren {e^{i \paren {\frac \pi 2 - \theta} } + e^{-i \paren {\frac \pi 2 - \theta} } }$ Cosine Exponential Formulation $\ds$ $=$ $\ds \frac 1 2 \paren {e^{i \frac \pi 2} e^{-i \theta} + e^{-i \frac \pi 2} e^{i \theta} }$ Exponential of Sum: Complex Numbers $\ds$ $=$ $\ds \frac 1 2 \paren {\paren {\map \cos {\frac \pi 2} + i \, \map \sin {\frac \pi 2} } e^{-i \theta} + \paren {\map \cos {-\frac \pi 2} + i \, \map \sin {-\frac \pi 2} } e^{i \theta} }$ Euler's Formula $\ds$ $=$ $\ds \frac 1 2 \paren {i e^{-i \theta} - i e^{i \theta} }$ Cosine of Right Angle, Sine of Right Angle, Cosine Function is Even, Sine Function is Odd $\ds$ $=$ $\ds \frac 1 {2 i} \paren {e^{i \theta} - e^{-i \theta} }$ Definition of Imaginary Unit $\ds$ $=$ $\ds \sin \theta$ Sine Exponential Formulation

$\blacksquare$

## Proof 4 Let $\angle xOP$ and $\angle QOy$ be complementary.

Then:

$\angle xOP = \angle QOy$

Hence:

the projection of $OP$ on the $x$-axis

equals:

the projection of $OQ$ on the $y$-axis.

Hence the result.

$\blacksquare$

## Historical Note

The result Cosine of Complement equals Sine was discovered and documented by Varahamihira in the $6$th century CE.